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A127922 1/24 of product of three numbers: n-th prime, previous and following number.

%I A127922 #55 Jan 20 2023 17:12:18
%S A127922 1,5,14,55,91,204,285,506,1015,1240,2109,2870,3311,4324,6201,8555,
%T A127922 9455,12529,14910,16206,20540,23821,29370,38024,42925,45526,51039,
%U A127922 53955,60116,85344,93665,107134,111895,137825,143450,161239,180441,194054
%N A127922 1/24 of product of three numbers: n-th prime, previous and following number.
%C A127922 The product of (n-1), n, and (n+1) = n^3 - n. - _Harvey P. Dale_, Jan 17 2011
%C A127922 For n > 2, a(n) = A001318(n-2) * A007310(n-1), if A007310(n-1) is prime. Also a(n) is a subsequence of A000330. - _Richard R. Forberg_, Dec 25 2013
%C A127922 If p is an odd prime it can always be the side length of a leg of a primitive Pythagorean triangle. However it constrains the other leg to have a side length of (p^2-1)/2 and the hypotenuse to have a side length of (p^2+1)/2. The resulting triangle has an area equal to (p-1)*p*(p+1)/4. a(n) is 1/6 the area of such triangles. - _Frank M Jackson_, Dec 06 2017
%H A127922 G. C. Greubel, <a href="/A127922/b127922.txt">Table of n, a(n) for n = 2..1000</a>
%H A127922 César Aguilera, <a href="https://hal.archives-ouvertes.fr/hal-02909691">Two Prime Number Objects and The Velucchi Numbers</a>, hal-02909691 [math.NT], 2020.
%F A127922 a(n) = A011842(A000040(n) + 1) = A000330((A000040(n) - 1) / 2).
%t A127922 Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/24, {n, 1, 100}] (#^3-#)/ 24&/@ Prime[Range[2,40]] (* _Harvey P. Dale_, Jan 17 2011 *)
%t A127922 ((#-1)#(#+1))/24&/@Prime[Range[2,40]] (* _Harvey P. Dale_, Jan 20 2023 *)
%o A127922 (PARI) for(n=2,25, print1((prime(n)+1)*prime(n)*(prime(n)-1)/24, ", ")) \\ _G. C. Greubel_, Jun 19 2017
%Y A127922 Cf. A000040, A000330, A011842.
%Y A127922 Cf. A036689, A034953, A127917, A127918, A127919, A127920, A127921, A011842.
%K A127922 nonn
%O A127922 2,2
%A A127922 _Artur Jasinski_, Feb 06 2007